Problem 805

Erdős Problem #805: declared status 'open'. Formalized: no. For which functions $g(n)$ with $n>g(n)\geq (\log n)^2$ is there a graph on $n$ vertices in which every induced subgraph on $g(n)$ vertices contains a clique of size $\geq \log n$ and an independent set of size $\geq \log n$? In particular, is there such a graph for $g(n)=(\log n)^3$? Prize: no. Tags: graph theory.